Most modern Solid State Model

I was curious as to what the most current model in solid state physics is. I've studied the Einstein and Debye Models in Stat Mech, as well as free electron and tight binding models in an intro solid state class, but these models are rather old(which does not imply they are useless). I am in a lab doing research as an undergrad, specifically dealing with semiconductors and phonons, and I wanted to learn about the more modern theories.

I think seeing the "state of the art" theory and then working backwards in time to a model which is simpler to work in and understand would be useful.

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I have found that DFT is a very popular way to do simulations involving solid state physics, however, from what I gather, it is not very "paper and pencil" friendly.

I'd really like to develop a better intuition for how phonons and electrons move through crystals(would it be approporiate to say transport properties?), as I tend to imagine both with the water and pipes model from intro E&M, which I do not think holds up well when dealing with real samples on the tens of nanometers scale.

I'm sorry for being long winded, so in short, I'd like to develop a better idea of what phonons and electrons "do" in a solid, and I think a model which is more advanced than Debye's or Einstein's may help/

The main issue with your first question is that there isn't really such a model. Or rather, one can only write something like [tex]H_{tot}=H_e + H_I + H_{int}[/tex] where [itex]H_{tot}[/itex] is the total Hamiltonian, [itex]H_{e}[/itex] is the electron Hamiltonian which contains the kinetic energy of each electron and all interactions between them. Similarly, [itex]H_I[/itex] contains the ions (usually thought of as vibrating around lattice points), while [itex]H_{int}[/itex] contains interactions betweens electrons and ions. This Hamiltonian contains everything (well, you need Schrödinger's equation too), and as the saying goes, then you just need to solve it. The problem is, of course, that the devil lies in the details.

For one, this model can't be solved in the general case, unlike the Debye and Einstein models. Instead one has to resort to various approximations or specializations. Often one simplifies the electron-ion interaction, for example, by assuming that each electron travels in a background potential from the other ions. DFT turns out to be a really good calculational tool (on computer, anyway), but it only works if you make certain assumptions. And, you guessed it, those assumptions are not true for all solid state systems!

What you can do, is to treat e.g. [itex]H_I[/itex] separately and assume that the interactions between the ions looks a bit like if each ion is a (quantum) harmonic oscillator and then look at how the wave solutions behave. Then assume that the electrons can be treated as independent, and that each electrons travels in a background potential due to the ions and the other electrons. It actually turns out that a periodic potential (like a lattice) makes the electrons move a bit like plane waves (see Bloch's theorem). By then it all starts reading like most introductory solid state books out there, so if you haven't read one yet, please start looking for one.

If you are thinking about a specific material, or samples of specific size, well know this: the details really matter. Is it a metal? A superconductor? A semiconductor? An insulator is different again. For small systems, there is always the question of boundary conditions. You might get some reflections of waves on the edges, and some propagation into nearby materials. Transport, by the way, usually means motion due to some external stimulus - maybe an electric field. Sometimes it is ballistic (when we can think of the electrons a bit like bullets) and at other times it will look more like diffusion. It really comes down to what choices and approximations one makes, so I really think your best way forward is to study some of the simpler examples from textbooks and then try to see parallels and differences to the samples you're working with yourself.

Hi Hypersphere, thank you for your response. I was hoping there was some kind of "midway" model between the very difficult models and Debye/Einstein. I guess I will just have to go crawl back to Kittel(ugh) and Ashcroft/Mermin.

If you were curious I'm looking into nanostructured thermoelectric materials, specifically on increaseing the thermoelectric figure of merit, ZT.